Bayesian data analysis demo 11.2

Metropolis algorithm

ggplot2 is used for plotting, tidyr for manipulating data frames

library(ggplot2)
theme_set(theme_minimal())
library(tidyr)
library(gganimate)
library(ggforce)
library(MASS)
library(rprojroot)
library(rstan)
root<-has_file(".BDA_R_demos_root")$make_fix_file()

Parameters of a normal distribution used as a toy target distribution

y1 <- 0
y2 <- 0
r <- 0.8
S <- diag(2)
S[1, 2] <- r
S[2, 1] <- r

Metropolis proposal distribution scale

sp <- 0.3

Sample from the toy distribution to visualize 90% HPD interval with ggplot's stat_ellipse()

dft <- data.frame(mvrnorm(100000, c(0, 0), S))

see BDA3 p. 85 for how to compute HPD for multivariate normal in 2d-case contour for 90% HPD is an ellipse, whose semimajor axes can be computed from the eigenvalues of the covariance matrix scaled by a value selected to get ellipse match the density at the edge of 90% HPD. Angle of the ellipse could be computed from the eigenvectors, but since the marginals are same we know that angle is pi/4 Starting value of the chain

t1 <- -2.5
t2 <- 2.5

Number of iterations.

M <- 5000

Insert your own Metropolis sampling here

# Allocate memory for the sample
tt <- matrix(rep(0, 2*M), ncol = 2)
tt[1,] <- c(t1, t2)    # Save starting point
# For demonstration load pre-computed values
# Replace this with your algorithm!
# tt is a M x 2 array, with M draws of both theta_1 and theta_2
load(root("demos_ch11","demo11_2a.RData"))

The rest is for illustration Take the first 200 draws to illustrate how the sampler works

df100 <- data.frame(id=rep(1,100),
                    iter=1:100, 
                    th1 = tt[1:100, 1],
                    th2 = tt[1:100, 2],
                    th1l = c(tt[1, 1], tt[1:(100-1), 1]),
                    th2l = c(tt[1, 2], tt[1:(100-1), 2]))

Take the first 5000 observations after warmup of 50

s <- 5000
warm <- 500
dfs <- data.frame(th1 = tt[(warm+1):s, 1], th2 = tt[(warm+1):s, 2])

Remove warm-up period of 50 first draws later

# labels and frame indices for the plot
labs1 <- c('Draws', 'Steps of the sampler', '90% HPD')
p1 <- ggplot() +
  geom_jitter(data = df100, width=0.05, height=0.05,
              aes(th1, th2, group=id, color ='1'), alpha=0.3) +
  geom_segment(data = df100, aes(x = th1, xend = th1l, color = '2',
                                 y = th2, yend = th2l)) +
  stat_ellipse(data = dft, aes(x = X1, y = X2, color = '3'), level = 0.9) +
  coord_cartesian(xlim = c(-4, 4), ylim = c(-4, 4)) +
  labs(x = 'theta1', y = 'theta2') +
  scale_color_manual(values = c('red', 'forestgreen','blue'), labels = labs1) +
  guides(color = guide_legend(override.aes = list(
    shape = c(16, NA, NA), linetype = c(0, 1, 1)))) +
  theme(legend.position = 'bottom', legend.title = element_blank())

The following generates a gif animation of the steps of the sampler (might take 10 seconds).

animate(p1 +   
          transition_reveal(along=iter) + 
          shadow_trail(0.01))

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Plot the final frame

p1

show 1000 draws after the warm-up

labs2 <- c('Draws', '90% HPD')
ggplot() +
  geom_point(data = dfs[1:1000,],
             aes(th1, th2, color = '1'), alpha = 0.3) +
  stat_ellipse(data = dft, aes(x = X1, y = X2, color = '2'), level = 0.9) +
  coord_cartesian(xlim = c(-4, 4), ylim = c(-4, 4)) +
  labs(x = 'theta1', y = 'theta2') +
  scale_color_manual(values = c('steelblue', 'blue'), labels = labs2) +
  guides(color = guide_legend(override.aes = list(
    shape = c(16, NA), linetype = c(0, 1), alpha = c(1, 1)))) +
  theme(legend.position = 'bottom', legend.title = element_blank())

show 4500 draws after the warm-up

labs2 <- c('Draws', '90% HPD')
ggplot() +
  geom_point(data = dfs,
             aes(th1, th2, color = '1'), alpha = 0.3) +
  stat_ellipse(data = dft, aes(x = X1, y = X2, color = '2'), level = 0.9) +
  coord_cartesian(xlim = c(-4, 4), ylim = c(-4, 4)) +
  labs(x = 'theta1', y = 'theta2') +
  scale_color_manual(values = c('steelblue', 'blue'), labels = labs2) +
  guides(color = guide_legend(override.aes = list(
    shape = c(16, NA), linetype = c(0, 1), alpha = c(1, 1)))) +
  theme(legend.position = 'bottom', legend.title = element_blank())

Convergence diagnostics

samp <- tt
dim(samp) <- c(dim(tt),1)
samp <- aperm(samp, c(1, 3, 2))
res<-monitor(samp, probs = c(0.25, 0.5, 0.75), digits_summary = 2)

## Inference for the input samples (1 chains: each with iter = 5000; warmup = 2500):
## 
##      Q5 Q50 Q95 Mean SD  Rhat Bulk_ESS Tail_ESS
## V1 -1.8   0 1.6 -0.1  1     1      101      152
## V2 -1.7   0 1.7  0.0  1     1       92      105
## 
## For each parameter, Bulk_ESS and Tail_ESS are crude measures of 
## effective sample size for bulk and tail quantities respectively (an ESS > 100 
## per chain is considered good), and Rhat is the potential scale reduction 
## factor on rank normalized split chains (at convergence, Rhat <= 1.05).

neff <- res[,'n_eff']
# both theta have owen neff, but for plotting these are so close to each
# other, so that single relative efficiency value is used
reff <- mean(neff/(s/2))

Visual convergence diagnostics

Collapse the data frame with row numbers augmented into key-value pairs for visualizing the chains

dfb <- dfs
sb <- s-warm
dfch <- within(dfb, iter <- 1:sb) %>% gather(grp, value, -iter)

Another data frame for visualizing the estimate of the autocorrelation function

nlags <- 50
dfa <- sapply(dfb, function(x) acf(x, lag.max = nlags, plot = F)$acf) %>%
  data.frame(iter = 0:(nlags)) %>% gather(grp, value, -iter)

A third data frame to visualize the cumulative averages and the 95% intervals

dfca <- (cumsum(dfb) / (1:sb)) %>%
  within({iter <- 1:sb
  uppi <-  1.96/sqrt(1:sb)
  upp <- 1.96/(sqrt(1:sb*reff))}) %>%
  gather(grp, value, -iter)

Visualize the chains

ggplot(data = dfch) +
  geom_line(aes(iter, value, color = grp)) +
  labs(title = 'Trends') +
  scale_color_discrete(labels = c('theta1','theta2')) +
  theme(legend.position = 'bottom', legend.title = element_blank())

Visualize the estimate of the autocorrelation function

ggplot(data = dfa) +
  geom_line(aes(iter, value, color = grp)) +
  geom_hline(aes(yintercept = 0)) +
  labs(title = 'Autocorrelation function') +
  scale_color_discrete(labels = c('theta1', 'theta2')) +
  theme(legend.position = 'bottom', legend.title = element_blank())

Visualize the estimate of the Monte Carlo error estimates

# labels
labs3 <- c('theta1', 'theta2',
           '95% interval for MCMC error',
           '95% interval for independent MC')
ggplot() +
  geom_line(data = dfca, aes(iter, value, color = grp, linetype = grp)) +
  geom_line(aes(1:sb, -1.96/sqrt(1:sb*reff)), linetype = 2) +
  geom_line(aes(1:sb, -1.96/sqrt(1:sb)), linetype = 3) +
  geom_hline(aes(yintercept = 0)) +
  coord_cartesian(ylim = c(-1.5, 1.5), xlim = c(0,4000)) +
  labs(title = 'Cumulative averages') +
  scale_color_manual(values = c('red','blue',rep('black', 2)), labels = labs3) +
  scale_linetype_manual(values = c(1, 1, 2, 3), labels = labs3) +
  theme(legend.position = 'bottom', legend.title = element_blank())

Same again with r=0.99 Parameters of a normal distribution used as a toy target distribution

y1 <- 0
y2 <- 0
r <- 0.99
S <- diag(2)
S[1, 2] <- r
S[2, 1] <- r

Metropolis proposal distribution scale

sp <- 0.3

Sample from the toy distribution to visualize 90% HPD interval with ggplot's stat_ellipse()

dft <- data.frame(mvrnorm(100000, c(0, 0), S))

see BDA3 p. 85 for how to compute HPD for multivariate normal in 2d-case contour for 90% HPD is an ellipse, whose semimajor axes can be computed from the eigenvalues of the covariance matrix scaled by a value selected to get ellipse match the density at the edge of 90% HPD. Angle of the ellipse could be computed from the eigenvectors, but since the marginals are same we know that angle is pi/4 Starting value of the chain

t1 <- -2.5
t2 <- 2.5

Number of iterations.

M <- 5000

Insert your own Metropolis sampling here

# Allocate memory for the sample
tt <- matrix(rep(0, 2*M), ncol = 2)
tt[1,] <- c(t1, t2)    # Save starting point
# For demonstration load pre-computed values
# Replace this with your algorithm!
# tt is a M x 2 array, with M draws of both theta_1 and theta_2
load(root("demos_ch11","demo11_2b.RData"))

The rest is for illustration Take the first 200 draws to illustrate how the sampler works

df100 <- data.frame(id=rep(1,100),
                    iter=1:100, 
                    th1 = tt[1:100, 1],
                    th2 = tt[1:100, 2],
                    th1l = c(tt[1, 1], tt[1:(100-1), 1]),
                    th2l = c(tt[1, 2], tt[1:(100-1), 2]))

Take the first 5000 observations after warmup of 50

s <- 5000
warm <- 500
dfs <- data.frame(th1 = tt[(warm+1):s, 1], th2 = tt[(warm+1):s, 2])

Remove warm-up period of 50 first draws later

# labels and frame indices for the plot
labs1 <- c('Draws', 'Steps of the sampler', '90% HPD')
p1 <- ggplot() +
  geom_jitter(data = df100, width=0.05, height=0.05,
             aes(th1, th2, group=id, color ='1'), alpha=0.3) +
  geom_segment(data = df100, aes(x = th1, xend = th1l, color = '2',
                                 y = th2, yend = th2l)) +
  stat_ellipse(data = dft, aes(x = X1, y = X2, color = '3'), level = 0.9) +
  coord_cartesian(xlim = c(-4, 4), ylim = c(-4, 4)) +
  labs(x = 'theta1', y = 'theta2') +
  scale_color_manual(values = c('red', 'forestgreen','blue'), labels = labs1) +
  guides(color = guide_legend(override.aes = list(
    shape = c(16, NA, NA), linetype = c(0, 1, 1)))) +
  theme(legend.position = 'bottom', legend.title = element_blank())

The following generates a gif animation of the steps of the sampler (might take 10 seconds).

animate(p1 +   
          transition_reveal(along=iter) + 
        shadow_trail(0.01))

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Plot the final frame

p1

show 1000 draws after the warm-up

labs2 <- c('Draws', '90% HPD')
ggplot() +
  geom_point(data = dfs[1:1000,],
             aes(th1, th2, color = '1'), alpha = 0.3) +
  stat_ellipse(data = dft, aes(x = X1, y = X2, color = '2'), level = 0.9) +
  coord_cartesian(xlim = c(-4, 4), ylim = c(-4, 4)) +
  labs(x = 'theta1', y = 'theta2') +
  scale_color_manual(values = c('steelblue', 'blue'), labels = labs2) +
  guides(color = guide_legend(override.aes = list(
    shape = c(16, NA), linetype = c(0, 1), alpha = c(1, 1)))) +
  theme(legend.position = 'bottom', legend.title = element_blank())

show 4500 draws after the warm-up

labs2 <- c('Draws', '90% HPD')
ggplot() +
  geom_point(data = dfs,
             aes(th1, th2, color = '1'), alpha = 0.3) +
  stat_ellipse(data = dft, aes(x = X1, y = X2, color = '2'), level = 0.9) +
  coord_cartesian(xlim = c(-4, 4), ylim = c(-4, 4)) +
  labs(x = 'theta1', y = 'theta2') +
  scale_color_manual(values = c('steelblue', 'blue'), labels = labs2) +
  guides(color = guide_legend(override.aes = list(
    shape = c(16, NA), linetype = c(0, 1), alpha = c(1, 1)))) +
  theme(legend.position = 'bottom', legend.title = element_blank())

Convergence diagnostics

samp <- tt
dim(samp) <- c(dim(tt),1)
samp <- aperm(samp, c(1, 3, 2))
res<-monitor(samp, probs = c(0.25, 0.5, 0.75), digits_summary = 2)

## Inference for the input samples (1 chains: each with iter = 5000; warmup = 2500):
## 
##      Q5  Q50 Q95 Mean  SD  Rhat Bulk_ESS Tail_ESS
## V1 -1.7 -0.4 1.5 -0.2 1.1  1.02       24       50
## V2 -1.6 -0.4 1.5 -0.2 1.1  1.03       24       51
## 
## For each parameter, Bulk_ESS and Tail_ESS are crude measures of 
## effective sample size for bulk and tail quantities respectively (an ESS > 100 
## per chain is considered good), and Rhat is the potential scale reduction 
## factor on rank normalized split chains (at convergence, Rhat <= 1.05).

neff <- res[,'n_eff']
# both theta have owen neff, but for plotting these are so close to each
# other, so that single relative efficiency value is used
reff <- mean(neff/(s/2))

Visual convergence diagnostics

Collapse the data frame with row numbers augmented into key-value pairs for visualizing the chains

dfb <- dfs
sb <- s-warm
dfch <- within(dfb, iter <- 1:sb) %>% gather(grp, value, -iter)

Another data frame for visualizing the estimate of the autocorrelation function

nlags <- 100
dfa <- sapply(dfb, function(x) acf(x, lag.max = nlags, plot = F)$acf) %>%
  data.frame(iter = 0:(nlags)) %>% gather(grp, value, -iter)

A third data frame to visualize the cumulative averages and the 95% intervals

dfca <- (cumsum(dfb) / (1:sb)) %>%
  within({iter <- 1:sb
          uppi <-  1.96/sqrt(1:sb)
          upp <- 1.96/(sqrt(1:sb*reff))}) %>%
  gather(grp, value, -iter)

Visualize the chains

ggplot(data = dfch) +
  geom_line(aes(iter, value, color = grp)) +
  labs(title = 'Trends') +
  scale_color_discrete(labels = c('theta1','theta2')) +
  theme(legend.position = 'bottom', legend.title = element_blank())

Visualize the estimate of the autocorrelation function

ggplot(data = dfa) +
  geom_line(aes(iter, value, color = grp)) +
  geom_hline(aes(yintercept = 0)) +
  labs(title = 'Autocorrelation function') +
  scale_color_discrete(labels = c('theta1', 'theta2')) +
  theme(legend.position = 'bottom', legend.title = element_blank())

Visualize the estimate of the Monte Carlo error estimates

# labels
labs3 <- c('theta1', 'theta2',
           '95% interval for MCMC error',
           '95% interval for independent MC')
ggplot() +
  geom_line(data = dfca, aes(iter, value, color = grp, linetype = grp)) +
  geom_line(aes(1:sb, -1.96/sqrt(1:sb*reff)), linetype = 2) +
  geom_line(aes(1:sb, -1.96/sqrt(1:sb)), linetype = 3) +
  geom_hline(aes(yintercept = 0)) +
  coord_cartesian(ylim = c(-1.5, 1.5), xlim = c(0,4000)) +
  labs(title = 'Cumulative averages') +
  scale_color_manual(values = c('red','blue',rep('black', 2)), labels = labs3) +
  scale_linetype_manual(values = c(1, 1, 2, 3), labels = labs3) +
  theme(legend.position = 'bottom', legend.title = element_blank())

Same again with sp = 1.5

sp = 1.5

Insert your own Metropolis sampling here

# Allocate memory for the sample
tt <- matrix(rep(0, 2*M), ncol = 2)
tt[1,] <- c(t1, t2)    # Save starting point
# For demonstration load pre-computed values
# Replace this with your algorithm!
# tt is a M x 2 array, with M draws of both theta_1 and theta_2
load(root("demos_ch11","demo11_2c.RData"))

The rest is for illustration Take the first 200 draws to illustrate how the sampler works

df100 <- data.frame(id=rep(1,100),
                    iter=1:100, 
                    th1 = tt[1:100, 1],
                    th2 = tt[1:100, 2],
                    th1l = c(tt[1, 1], tt[1:(100-1), 1]),
                    th2l = c(tt[1, 2], tt[1:(100-1), 2]))

Take the first 5000 observations after warmup of 50

s <- 5000
warm <- 500
dfs <- data.frame(th1 = tt[(warm+1):s, 1], th2 = tt[(warm+1):s, 2])

Remove warm-up period of 50 first draws later

# labels and frame indices for the plot
labs1 <- c('Draws', 'Steps of the sampler', '90% HPD')
p1 <- ggplot() +
  geom_jitter(data = df100, width=0.05, height=0.05,
             aes(th1, th2, group=id, color ='1'), alpha=0.3) +
  geom_segment(data = df100, aes(x = th1, xend = th1l, color = '2',
                                 y = th2, yend = th2l)) +
  stat_ellipse(data = dft, aes(x = X1, y = X2, color = '3'), level = 0.9) +
  coord_cartesian(xlim = c(-4, 4), ylim = c(-4, 4)) +
  labs(x = 'theta1', y = 'theta2') +
  scale_color_manual(values = c('red', 'forestgreen','blue'), labels = labs1) +
  guides(color = guide_legend(override.aes = list(
    shape = c(16, NA, NA), linetype = c(0, 1, 1)))) +
  theme(legend.position = 'bottom', legend.title = element_blank())

The following generates a gif animation of the steps of the sampler (might take 10 seconds).

animate(p1 +   
          transition_reveal(along=iter) + 
        shadow_trail(0.01))

## Rendering [--------------------------------------------] at 3.3 fps ~ eta: 30s
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show 1000 draws after the warm-up

labs2 <- c('Draws', '90% HPD')
ggplot() +
  geom_point(data = dfs[1:1000,],
             aes(th1, th2, color = '1'), alpha = 0.3) +
  stat_ellipse(data = dft, aes(x = X1, y = X2, color = '2'), level = 0.9) +
  coord_cartesian(xlim = c(-4, 4), ylim = c(-4, 4)) +
  labs(x = 'theta1', y = 'theta2') +
  scale_color_manual(values = c('steelblue', 'blue'), labels = labs2) +
  guides(color = guide_legend(override.aes = list(
    shape = c(16, NA), linetype = c(0, 1), alpha = c(1, 1)))) +
  theme(legend.position = 'bottom', legend.title = element_blank())

show 4500 draws after the warm-up

labs2 <- c('Draws', '90% HPD')
ggplot() +
  geom_point(data = dfs,
             aes(th1, th2, color = '1'), alpha = 0.3) +
  stat_ellipse(data = dft, aes(x = X1, y = X2, color = '2'), level = 0.9) +
  coord_cartesian(xlim = c(-4, 4), ylim = c(-4, 4)) +
  labs(x = 'theta1', y = 'theta2') +
  scale_color_manual(values = c('steelblue', 'blue'), labels = labs2) +
  guides(color = guide_legend(override.aes = list(
    shape = c(16, NA), linetype = c(0, 1), alpha = c(1, 1)))) +
  theme(legend.position = 'bottom', legend.title = element_blank())

Convergence diagnostics

samp <- tt
dim(samp) <- c(dim(tt),1)
samp <- aperm(samp, c(1, 3, 2))
res<-monitor(samp, probs = c(0.25, 0.5, 0.75), digits_summary = 2)

## Inference for the input samples (1 chains: each with iter = 5000; warmup = 2500):
## 
##      Q5  Q50 Q95 Mean  SD  Rhat Bulk_ESS Tail_ESS
## V1 -1.8 -0.5 0.8 -0.5 0.8  1.00       45       34
## V2 -1.9 -0.4 0.8 -0.5 0.8  1.01       45       33
## 
## For each parameter, Bulk_ESS and Tail_ESS are crude measures of 
## effective sample size for bulk and tail quantities respectively (an ESS > 100 
## per chain is considered good), and Rhat is the potential scale reduction 
## factor on rank normalized split chains (at convergence, Rhat <= 1.05).

neff <- res[,'n_eff']
# both theta have owen neff, but for plotting these are so close to each
# other, so that single relative efficiency value is used
reff <- mean(neff/(s/2))

Visual convergence diagnostics

Collapse the data frame with row numbers augmented into key-value pairs for visualizing the chains

dfb <- dfs
sb <- s-warm
dfch <- within(dfb, iter <- 1:sb) %>% gather(grp, value, -iter)

Another data frame for visualizing the estimate of the autocorrelation function

nlags <- 100
dfa <- sapply(dfb, function(x) acf(x, lag.max = nlags, plot = F)$acf) %>%
  data.frame(iter = 0:(nlags)) %>% gather(grp, value, -iter)

A third data frame to visualize the cumulative averages and the 95% intervals

dfca <- (cumsum(dfb) / (1:sb)) %>%
  within({iter <- 1:sb
          uppi <-  1.96/sqrt(1:sb)
          upp <- 1.96/(sqrt(1:sb*reff))}) %>%
  gather(grp, value, -iter)

Visualize the chains

ggplot(data = dfch) +
  geom_line(aes(iter, value, color = grp)) +
  labs(title = 'Trends') +
  scale_color_discrete(labels = c('theta1','theta2')) +
  theme(legend.position = 'bottom', legend.title = element_blank())

Visualize the estimate of the autocorrelation function

ggplot(data = dfa) +
  geom_line(aes(iter, value, color = grp)) +
  geom_hline(aes(yintercept = 0)) +
  labs(title = 'Autocorrelation function') +
  scale_color_discrete(labels = c('theta1', 'theta2')) +
  theme(legend.position = 'bottom', legend.title = element_blank())

Visualize the estimate of the Monte Carlo error estimates

# labels
labs3 <- c('theta1', 'theta2',
           '95% interval for MCMC error',
           '95% interval for independent MC')
ggplot() +
  geom_line(data = dfca, aes(iter, value, color = grp, linetype = grp)) +
  geom_line(aes(1:sb, -1.96/sqrt(1:sb*reff)), linetype = 2) +
  geom_line(aes(1:sb, -1.96/sqrt(1:sb)), linetype = 3) +
  geom_hline(aes(yintercept = 0)) +
  coord_cartesian(ylim = c(-1.5, 1.5), xlim = c(0,4000)) +
  labs(title = 'Cumulative averages') +
  scale_color_manual(values = c('red','blue',rep('black', 2)), labels = labs3) +
  scale_linetype_manual(values = c(1, 1, 2, 3), labels = labs3) +
  theme(legend.position = 'bottom', legend.title = element_blank())